Optimal estimator for the survival distribution and related quantities for treatment policies in two-stage randomization designs in clinical trials

Two-stage designs, where patients are initially randomized to an induction therapy and then depending upon their response and consent; are randomized to a maintenance therapy, are common in cancer and other clinical trials. The goal is to compare different combinations of primary and maintenance therapies to find the combination that is most beneficial. In practice, the analysis is usually conducted in two separate stages which does not directly address the major objective of finding the best combination. Recently Lunceford, Davidian, and Tsiatis (2002, Biometrics 58, 48-57) introduced ad hoc estimators for the survival distribution and mean restricted survival tine under different treatment policies. These estimators are consistent but not efficient; and do not include information from auxiliary covariates. In this article we derive estimators that are easy to compute and are more efficient than previous estimators. We also show how to improve efficiency further by taking into account additional information from auxiliary variables. Large sample properties of these estimators are derived and comparisons with other estimators are made using simulation. We apply our estimators to a leukemia clinical trial data set that motivated this study.